Abstract

Observations and GCMs exhibit approximate proportionality between cumulative carbon dioxide (CO\(_{2}\)) emissions and global warming. Here we identify sufficient conditions for the relationship between cumulative CO\(_{2}\) emissions and global warming to be independent of the path of CO\(_{2}\) emissions; referred to as “path independence”. Our starting point is a closed form expression for global warming in a two-box energy balance model (EBM), which depends explicitly on cumulative emissions, airborne fraction and time. Path independence requires that this function can be approximated as depending on cumulative emissions alone. We show that path independence arises from weak constraints, occurring if the timescale for changes in cumulative emissions (equal to ratio between cumulative emissions and emissions rate) is small compared to the timescale for changes in airborne fraction (which depends on CO\(_{2}\) uptake), and also small relative to a derived climate model parameter called the damping-timescale, which is related to the rate at which deep-ocean warming affects global warming. Effects of uncertainties in the climate model and carbon cycle are examined. Large deep-ocean heat capacity in the Earth system is not necessary for path independence, which appears resilient to climate modeling uncertainties. However long time-constants in the Earth system carbon cycle are essential, ensuring that airborne fraction changes slowly with timescale much longer than the timescale for changes in cumulative emissions. Therefore path independence between cumulative emissions and warming cannot arise for short-lived greenhouse gases.

For this error to be significant the timescale \(\varsigma _{r}\), indicating how rapidly airborne fraction changes, must not be small compared to timescale \(\varsigma _{m1}\). This corresponds to rapid changes in the airborne fraction. From equation (3) the ratio of timescales is

where \(\tau _{m}=m/\dot{m}\) is the timescale for the rate of change of emissions, as defined in Sect. 3.1. The above ratio, and corresponding error in neglecting the partial effect of airborne fraction, would not be small if there are rapid changes in the emissions rate, especially after the airborne fraction has become small; or if emissions are small, because of the last term above.

Error \(\Delta u_{t}\) from neglecting partial effect of time

From Sect. 3.3, the ratio of the partial effect of time and the partial effect of cumulative emissions is

Here we demonstrate that global warming is a concave function of cumulative emissions in the linear two-box EBM, in contrast to GCMs where it is approximately linear (Tokarska et al. (2016)). Assuming that sufficient conditions for path independence are met so that \(\varsigma _{m1}\ll \varsigma _{r}\) and \(\varsigma _{m1}\ll \tau _{D}\) from Eq. (13), we consider the departure from linearity in this simple model. Then

and the departure from linearity between cumulative CO\(_{2}\) emissions and global warming is related to second derivative \(d^{2}u/dm_{1}^{2}\). This depends on how much the increase in the second term above can compensate for the decrease in r/g , as cumulative emissions grows and airborne fraction decreases. Applying the differential operator in Eq. (9) and computing the ratio of 2nd and 1st—degree contributions to series expansion of u about a given value of cumulative emissions, the relative departure from linearity is measured by

which is negative, and global warming is generally a concave function of cumulative emissions of CO\(_{2}\) in this EBM (Figs. 6, 7, 8). Please refer to SI for further details; mixed derivatives \(\partial ^{2}u/\partial r\partial m_{1}\) and \(\partial ^{2}u/\partial t\partial m_{1}\) are small leading to the simplification above.

The most important physical parameter for the above ratio is damping-timescale \(\tau _{D}\). We have \(\frac{g^{2}}{r^{2}}e{}^{-t/\tau _{l}}\int _{0}^{t}e{}^{z/\tau _{l}}\frac{r^{2}}{g^{2}}dz>\frac{g}{r}e{}^{-t/\tau _{l}}\int _{0}^{t}e{}^{z/\tau _{l}}\frac{r}{g}dz\), because \(\frac{r}{g}\) is decreasing, and therefore the above ratio increases in magnitude with \(1/\tau _{D}\).11 In case \(\tau _{D}\) is longer, the relation between cumulative CO\(_{2}\) emissions and global warming is closer to linearity.